Question

∠uxy and ∠vux are supplementary. complete the proof that \overleftrightarrow{wy} \parallel \overleftrightarrow{tv}.

statement	reason
2 m∠uxy + m∠vux = 180°
3 m∠suv + m∠vux = 180°
4 m∠uxy + m∠vux = m∠suv + m∠vux
5 m∠uxy = m∠suv
6 \overleftrightarrow{wy} \parallel \overleftrightarrow{tv}

Explanation:

Step1: Given

\(\angle UXY\) and \(\angle VUX\) are supplementary (given in the problem statement).

Step2: Definition of supplementary angles

If two angles are supplementary, then the sum of their measures is \(180^{\circ}\). So, \(m\angle UXY + m\angle VUX=180^{\circ}\).

Step3: Linear - pair postulate

\(\angle SUV\) and \(\angle VUX\) form a linear pair. By the linear - pair postulate, \(m\angle SUV + m\angle VUX = 180^{\circ}\).

Step4: Transitive property of equality

Since \(m\angle UXY + m\angle VUX = 180^{\circ}\) (from Step 2) and \(m\angle SUV + m\angle VUX=180^{\circ}\) (from Step 3), we can use the transitive property of equality (\(a = c\) and \(b = c\) implies \(a = b\)). So, \(m\angle UXY + m\angle VUX=m\angle SUV + m\angle VUX\).

Step5: Subtraction property of equality

Subtract \(m\angle VUX\) from both sides of the equation \(m\angle UXY + m\angle VUX=m\angle SUV + m\angle VUX\). We get \(m\angle UXY=m\angle SUV\).

Step6: Alternate - exterior angles converse

If two lines are cut by a transversal and the alternate - exterior angles are congruent, then the two lines are parallel. Here, \(\overleftrightarrow{WY}\) and \(\overleftrightarrow{TV}\) are cut by transversal \(\overleftrightarrow{SZ}\), and \(\angle UXY\) and \(\angle SUV\) are alternate - exterior angles. Since \(m\angle UXY=m\angle SUV\), by the alternate - exterior angles converse, \(\overleftrightarrow{WY}\parallel\overleftrightarrow{TV}\).

Answer:

1. Given
2. Definition of supplementary angles
3. Linear - pair postulate
4. Transitive property of equality
5. Subtraction property of equality
6. Alternate - exterior angles converse